Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter12: Angle Relationships And Transformations
Section12.5: Reflections And Symmetry
Problem 20E
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![### Geometry Problem: Rhombus Angle Measurement
**Question:**
What is the measure of ∠1 in rhombus ABCD?
**Diagram Description:**
In the given diagram, there is a rhombus labeled ABCD. The diagonals of the rhombus intersect at point O, forming four angles at the center. The diagonals bisect each other, and the properties of a rhombus dictate that these diagonals split the rhombus's angles into two equal halves.
The labeling of the angles around point O is provided as follows:
- ∠AOB is labeled as 1
- ∠AOD is labeled as 2
- ∠COD is labeled as 3
- ∠BOC is labeled as 4
Additionally, one of the angles, ∠ADC, is provided as 53°.
**Multiple Choice Options:**
- 53°
- 90°
- 106°
- 180°
**Explanation of the Problem:**
In a rhombus, the diagonals bisect the angles at the vertices and are also perpendicular bisectors of each other. Given that ∠ADC is 53°, the diagonal AD bisects this angle, forming two equal angles at point O.
Since the diagonals of a rhombus bisect the internal angles, each part of the bisected angle is half of the original. Therefore, ∠AOD and ∠COD each are:
\[
\angle AOD = \angle COD = \frac{53°}{2} = 26.5°
\]
It's a property of the rhombus's diagonals to meet at right angles (90°). Thus, ∠AOB, labeled as ∠1, being one of the angles formed by the intersection of the diagonals, should be 90°.
**Answer:**
The measure of ∠1 is **90°**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47897a37-2eac-43d5-a731-3aff060a1d87%2Fe42511c7-b92a-4c5c-a2a7-8a5d729b5600%2Fpa5dcgl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem: Rhombus Angle Measurement
**Question:**
What is the measure of ∠1 in rhombus ABCD?
**Diagram Description:**
In the given diagram, there is a rhombus labeled ABCD. The diagonals of the rhombus intersect at point O, forming four angles at the center. The diagonals bisect each other, and the properties of a rhombus dictate that these diagonals split the rhombus's angles into two equal halves.
The labeling of the angles around point O is provided as follows:
- ∠AOB is labeled as 1
- ∠AOD is labeled as 2
- ∠COD is labeled as 3
- ∠BOC is labeled as 4
Additionally, one of the angles, ∠ADC, is provided as 53°.
**Multiple Choice Options:**
- 53°
- 90°
- 106°
- 180°
**Explanation of the Problem:**
In a rhombus, the diagonals bisect the angles at the vertices and are also perpendicular bisectors of each other. Given that ∠ADC is 53°, the diagonal AD bisects this angle, forming two equal angles at point O.
Since the diagonals of a rhombus bisect the internal angles, each part of the bisected angle is half of the original. Therefore, ∠AOD and ∠COD each are:
\[
\angle AOD = \angle COD = \frac{53°}{2} = 26.5°
\]
It's a property of the rhombus's diagonals to meet at right angles (90°). Thus, ∠AOB, labeled as ∠1, being one of the angles formed by the intersection of the diagonals, should be 90°.
**Answer:**
The measure of ∠1 is **90°**.
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